2,183 research outputs found
Lagrangian-Eulerian Methods for Uniqueness in Hydrodynamic Systems
We present a Lagrangian-Eulerian strategy for proving uniqueness and local
existence of solutions in path spaces of limited smoothness for a class of
incompressible hydrodynamic models including Oldroyd-B type complex fluid
models and zero magnetic resistivity magneto-hydrodynamics equations
Transport in Rotating Fluids
We consider uniformly rotating incompressible Euler and Navier-Stokes
equations. We study the suppression of vertical gradients of Lagrangian
displacement ("vertical" refers to the direction of the rotation axis). We
employ a formalism that relates the total vorticity to the gradient of the
back-to-labels map (the inverse Lagrangian map, for inviscid flows, a diffusive
analogue for viscous flows). The results include a nonlinear version of the
Taylor-Proudman theorem: in a steady solution of the rotating Euler equations,
two fluid material points which were initially on a vertical vortex line, will
perpetually maintain their vertical separation unchanged. For more general
situations, including unsteady flows, we obtain bounds for the vertical
gradients of the Lagrangian displacement that vanish linearly with the maximal
local Rossby number
Remarks on the fractional Laplacian with Dirichlet boundary conditions and applications
We prove nonlinear lower bounds and commutator estimates for the Dirichlet
fractional Laplacian in bounded domains. The applications include bounds for
linear drift-diffusion equations with nonlocal dissipation and global existence
of weak solutions of critical surface quasi-geostrophic equations
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